Theoretically, in addition to \(R(t)\) , it has also been possible to obtain the spatio-temporal behavior of different thermodynamic quantities in the hydrodynamic regime. The variation of density, pressure, temperature, and velocity with spatial distance, \(r\) , and time, \(t\) , are described by continuity equations for mass, momentum, and energy. These hydrodynamic equations correspond to the Navier Stokes equation. In the scaling limit \(r \rightarrow \infty \) , \(t \rightarrow \infty \)  keeping \(r t^{-2/(d+2)}\)  fixed, and after appropriately non-dimensionalizing the thermodynamic quantities, it was shown that the heat conduction and viscosity terms vanish and hence can be dropped. The Euler equations, resulting from dropping heat conduction and viscosity terms, in three dimensions were solved exactly by Taylor, von-Neumann and Sedov [1]}, [2]}, [3]}, [4]}, [5]} to obtain the scaling functions analytically, and we will refer to these self-similar solutions as the TvNS solution. The significance of the TvNS solution is in its wide applicability. For instance, it has been used for modeling evolution of supernova remnants in the adiabatic stage [6]}, [7]}, [8]}, [9]}, [10]}, [11]}. TvNS theory has been used to study systems where energy is continuously input in a localized region of space [12]}, [13]}.
TvNS theory has also been generalized for granular systems, where conserved quantities are fewer in number because energy is no longer conserved [14]}, [15]}. There are many examples in granular systems where a blast wave is generated when perturbed by either a single impact or by continuous energy injection. For instance, crater formation in granular bed due to single impact of an object or continuous jet [16]}, [17]}, [18]}, shock propagation in granular medium due to sudden impact [19]}, [20]}, [21]}, viscous fingering due to continuous energy injection [22]}, [23]}, [24]}, [25]}, [26]} or shock propagation in continuously driven granular media [27]}.
